#72209759

order	1536
automorphism	?
homology	Abelianization (H₁) = C₂²×C₃ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Splittings

Fitting, Sylow	#156685 ⋊ C₃
derived	C₂⁴×Q₈ ⋊ C₂²×C₃

Subgroups

	subgroup	G/H
center, hypercenter	C₂	#11709314
commutator, cosocle, hypocenter	C₂⁴×Q₈	C₂²×C₃
Frattini, socle	C₂⁵	C₂²×A₄
Fitting	#156685	C₃
LCS	G ▷ C₂⁴×Q₈
UCS	C₁ ◁ C₂

4097 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 4097 conjugacy classes of subgroups — too many to draw. Only its 47 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

Polycyclic presentation:

2^2,3,2^7|d^2=j,e^2=j,d^a=di,d^b=dfg,d^c=e,e^a=eh,e^b=ef,e^c=dej,e^d=ej,f^c=g,g^c=fg,h^c=hi,i^c=h

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72209764
maximal normal subgroup of	#72209764
maximal quotient of	#72227305

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 3, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 5, F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7);;
gap> SetConjugate(coll, 4, 3, F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.6);;
gap> SetConjugate(coll, 5, 3, F.4*F.5*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 3, F.7);;
gap> SetConjugate(coll, 7, 3, F.6*F.7);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 9, 3, F.8);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);